Finite transfer systems for general prime moduli and alphabets

Prove that, for every prime p and finite alphabet I, the first-passage kernel φ(v,z) modulo p can be computed by a finite F_p-weighted transfer system reading base-p digit columns, and establish a state-count bound in terms of p and |I| to resolve the fair-assignment cost for all prime moduli.

Background

For the binary alphabet with modulus 2, the paper constructs a seven-state automaton that computes the mod-2 first-passage kernel and reduces exact group assignment to O(T log T) bit operations. Open Problem P1 asks whether an analogous finite transfer system exists for every prime modulus and finite alphabet.

The proposed generalization uses Lucas multinomial weights, carry-debt kernels, and delta-type states. The paper explains that terminal full blocks should contribute zero modulo p through a free cyclic action, but a proof of finite-state closure and a bound on the number of states remain unresolved. For composite moduli, the problem additionally involves the interaction among the prime-power stopping sets and Kummer’s multi-carry criterion.

References

It is conjectured that for every prime $p$ and finite alphabet $I$, the kernel $\varphi(v,z) \bmod p$ can be computed by a finite $F_p$-weighted transfer system that reads base-$p$ digit columns, utilizing Lucas multinomial weights and two types of states, namely carry-debt kernels $\mathbf{1}[v \in C]\,\varphi(v + w, \cdot)$ for debt vectors $w$ in a bounded set, and $\delta$-type states carrying $H_p$/$C$ certificates.

— Algorithms, Complexity, and Entropy of the Bernard-Letac Fair-Sampling Construction  (2608.20234 - Gravel, 20 Aug 2026) in Open Problem P1, Section 5; discussed also in Section 4, subsection “Statement of the transfer theorem”